On Lipschitzian operators of substitution generated by set-valued functions

Abstract

We consider the Nemytskii operator, i.e., the operator of substitution, defined by $(N \phi)(x):=G(x,\phi(x))$, where $G$ is a given multifunction. It is shown that if $N$ maps a Hölder space $H_{\alpha}$ into $H_{\beta}$ and $N$ fulfils the Lipschitz condition then $G(x,y)=A(x,y)+B(x), (1)$ where $A(x,\cdot)$ is linear and $A(\cdot ,y),\, B \in H_{\beta}$. Moreover, some conditions are given under which the Nemytskii operator generated by $(1)$ maps $H_{\alpha}$ into $H_{\beta}$ and is Lipschitzian.